This part looks very interesting.
In particular, . Had the author of my book showed it as you did, I would not have trouble understanding it.
What are the and ?

are jus arbitrary subsets of the naturals. We are trying to show that if two subsets of the naturals aren't equal that their image under isn't equal. I jused to mean the same thing as but to indicate that it was for and not .

is just some point that is in or (I assumed WLOG above that ) which isn't in the other. We know there must be at least one since the two sets aren't equal.